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CRC Press eBooks

Uncertainty Principle

F.J. Duarte

Published Oct 29, 2013• 0 citations
Publisher

Abstract

R. P. Feynman, by far, physics’ most successful communicator, said: The uncertainty principle protects quantum mechanics. W. Heisenberg recognized that if it were possible to measure the momentum and the position simultaneously with a greater accuracy, quantum mechanics would collapse. Quantum mechanics maintains its perilous but correct existence”. In his book on Quantum Mechanics and Path Integrals , Feynman makes use of the two-slit experiment to provide an approximate description of the physics behind the uncertainty principle. Feynman’s description is outlined using a slightly different notation. The uncertainty principle is widely applied in optics. It applies to interferometry, linewidth measurements, and beam divergence measurements. The chapter focuses on the uncertainty principle and beam divergence. An important application of the uncertainty principle manifests itself in calculations of the angular resolution limit of telescopes used in astronomical observations. The Heisenberg uncertainty principle can be used to derive some useful identities in optics and interferometry.

DOI
10.1201/b16055-3

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R. P. Feynman, by far, physics’ most successful communicator, said: The uncertainty principle protects quantum mechanics. W. Heisenberg recognized that if it were possible to measure the momentum and the position simultaneously with a greater accuracy, quantum mechanics would collapse. Quantum mechanics maintains its perilous but correct existence”. In his book on Quantum Mechanics and Path Integrals , Feynman makes use of the two-slit experiment to provide an approximate description of the physics behind the uncertainty principle. Feynman’s description is outlined using a slightly different notation. The uncertainty principle is widely applied in optics. It applies to interferometry, linewidth measurements, and beam divergence measurements. The chapter focuses on the uncertainty principle and beam divergence. An important application of the uncertainty principle manifests itself in calculations of the angular resolution limit of telescopes used in astronomical observations. The Heisenberg uncertainty principle can be used to derive some useful identities in optics and interferometry.
One of the most striking deviations of quantum mechanics from classical mechanics is that any measurement of a microscopic system involves the interaction with the measuring instruments. The interaction cannot be neglected and in fact leads to uncontrollable disturbance on the system. In 1927, Werner Karl Heisenberg in his celebrated paper established the quantitative expression for unavoidable momentum disturbance caused by any position measurement. The crucial word here is simultaneously. The uncertainty relation gives the lower bound on the product of the uncertainties of two pairs of variables. But it says nothing about the accuracy with which individual observables can be measured. The physical meaning of energy-time uncertainty is different from that of x-px uncertainty. Position and momentum are observables to which Hermitian operators are assigned and which are measured at a particular time. An important implication of the uncertainty principle is that quantum systems do follow classical trajectories which conflict with our everyday experience.
Until Heisenberg proposed the uncertainty principle in 1927, it was an open question whether models for the structure of matter reflected actual reality or were useful fictions for operational purposes. There wasn’t any sense in either case that there might be parameters it was impossible to determine. The uncertainty principle says that it is impossible to determine simultaneously the location and momentum of an electron. While it applies to a specific situation, ‘uncertainty’ is a good description for the entire approach of quantum theory. Schrödinger’s equations show that electrons behave simultaneously as waves and particles. The double-slit experiment, when a beam of light or electrons passes through two parallel slits, shows that the act of observation influences the behavior of the wave–particle duality. The ability of a pair of ‘entangled’ electrons (or other particles) to influence each other instantaneously appears to exceed the speed limit of light. Paradoxes of quantum behavior may even extend to biological situations. Intuition or common sense are no guide to quantum behavior.
This paper is a small discussion about the uncertainty principle. With the great discoveries and researches made by predecessors, I try to answer some questions within the uncertainty principle. Using the definition of momentum, the momentum in de Broglie wave, the property of cosine function, the Euler’s formula, the basic property of plane wave, the law of derivative, the energy of quantum discovered by Max Planck, the photon energy discovered by Albert Einstein, the energy equation of a system and the Hamiltonian operator, I figure out the Schrodinger equation successfully. With the help of Schrodinger equation, the possibility density of a wave function, the concept of normalization, the average value of position and momentum, the Fourier transform, the integration by parts and the limit of normalized wave functions, I find the position operator and the momentum operator. Then using some knowledge of the standard deviation and the variance, the concept of inner product of vectors, some property of Dirac symbol, the law of calculation of complex number and the law of operator calculation, I obtain the general uncertainty relationship of abstract mathematical quantities. According to the general relationship, I get the inequality of uncertainty principle, and discuss the specific relationship between the position and the momentum. After getting the inequality, I use Heisenberg’s experiment of measuring an electron by using the γ ray to hit the electron to explain why we cannot measure the position and the momentum simultaneously. Based on Heisenberg’s experiment, I give my own explanation by discussing the earth and a spacecraft. Transforming the perspective, we can easily see how the accuracies of position and momentum change. In addition, I attempt to figure out why the uncertainty exists. However, I can only explain it by discussing the time and frequency in mathematical method and have no idea to work it out in a pure physics approach.
Most physics books do not reveal clearly how the Heisenberg uncertainty principle was derived. This uncertainty comes from the consequence of the wave-particle duality of matter giving statement that position and momentum cannot be measured in the same time. This article tries to reveal mathematics background behind the expression the Heisenberg uncertainty using supported mathematics background such as Fourier transform, Fourier transform integral, the probability of Gaussian distribution and it ends up with the expression of wave function which describe the localized particle giving relation Heisenberg uncertainty principle.
As a matter of fact, the Heisenberg’s uncertainty principle has important significance and extensive application in microcosmic particle science, is one of the cornerstones of quantum mechanics, proposed by Heisenberg in 1927. The uncertainty principle means that it is impossible to measure a particle’s speed and position at the same time. The uncertainty of particle, i.e., the product of uncertainty of position (∆Q) and uncertainty of momentum (∆P), is inevitable greater than or equal to Plank’s constant, it expresses the inconsistency between particle behavior in microcosm and macroscopic substance. This study is going to set forth the uncertainty meaning, the research progress of uncertainty principle from the angle of physics, and further exploring the physical significance of Heisenberg’s uncertainty principle. According to the analysis, the state-of-art applications will be demonstrated as well as the current limitations and prospects will be discussed. Overall, these results pave a novel path for further investigation in regarding to Heisenberg’s uncertainty.